Solution.

ABCD
From either extremity of the given right line
A B
draw a line
A C
perpendicular
(XI),
and equal to it
(III);
through
C
draw
C D
parallel to
A B(XXXI),
and through
B
draw
B D
parallel to
A C;
A D
is the required square.

Demonstration.

Because
A D
is a parallelogram (const.), and the angle
A
a right angle, the angles
C,
D,
and
B
are also right
(153);
and because
A C
is equal to
A B
(const.), and the sides
C D
and
D B
are equal to
A B
and
A C(XXXIV),
the four sides
A B,
A C,
C D,
D B
are equal, therefore
A D
is a square.